Exercising the right to remain silent may be used to protect a person from?

By exercising your right to remain silent you are protecting yourself from offering information that may be used as evidence against you in a criminal trial. You may also be protecting yourself from providing an untruthful alibi, the evidence of which may also be used against you in a criminal trial.

However, your "right to remain silent" is only applicable when you are in custody and the police are interrogating you. (see "Miranda v. Arizona")

You always have the right to remain silent, even when you are not under arrest. You are never compelled to testify against yourself.

You can never be physically compelled to answer a question, of course, but your constitutional right to avoid providing incriminating testimony is only present when you are in custody and are being interrogated. In that circumstance there cannot be any negative consequences if you refuse to answer questions. However, in other circumstances, you can be arrested for interfering with an officer if you refuse to answer basic questions or identify yourself. An example would be if you are stopped on the street by an officer investigating a crime for which you fit the description of the perpetrator and you refuse to identify yourself. In that circumstance you could be arrested.